To calculate a 95% confidence interval for the population mean \(\mu\) based on the given sample data, we can follow these steps:
- Calculate the sample mean (\(\bar{x}\)).
- Calculate the sample standard deviation (\(s\)).
- Determine the sample size (\(n\)).
- Determine the critical value (\(t^*\)) for a 95% confidence interval using the t-distribution.
- Calculate the margin of error.
- Construct the confidence interval.
Let's go through each step.
Step 1: Calculate the sample mean
\[ \bar{x} = \frac{\sum x_i}{n} \]
Given the sample data:
- \(x_1 = 36.1\)
- \(x_2 = 89.4\)
- \(x_3 = 69.1\)
- \(x_4 = 61.9\)
- \(x_5 = 104.2\)
- \(x_6 = 41.4\)
- \(x_7 = 67.1\)
- \(x_8 = 54.6\)
- \(x_9 = 90.2\)
- \(x_{10} = 68.9\)
Calculating the total sum:
\[ \sum x_i = 36.1 + 89.4 + 69.1 + 61.9 + 104.2 + 41.4 + 67.1 + 54.6 + 90.2 + 68.9 = 683.8 \]
The sample size \(n\) is 10.
Thus, the sample mean is:
\[ \bar{x} = \frac{683.8}{10} = 68.38 \]
Step 2: Calculate the sample standard deviation
The formula for the sample standard deviation is:
\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \]
Calculating \( (x_i - \bar{x})^2 \):
- \( (36.1 - 68.38)^2 = 1032.0644 \)
- \( (89.4 - 68.38)^2 = 444.0644 \)
- \( (69.1 - 68.38)^2 = 0.5184 \)
- \( (61.9 - 68.38)^2 = 41.5844 \)
- \( (104.2 - 68.38)^2 = 1294.9444 \)
- \( (41.4 - 68.38)^2 = 724.0644 \)
- \( (67.1 - 68.38)^2 = 1.6324 \)
- \( (54.6 - 68.38)^2 = 188.0644 \)
- \( (90.2 - 68.38)^2 = 473.1844 \)
- \( (68.9 - 68.38)^2 = 0.2724 \)
Now summing those squared differences:
\[ \sum (x_i - \bar{x})^2 = 1032.0644 + 444.0644 + 0.5184 + 41.5844 + 1294.9444 + 724.0644 + 1.6324 + 188.0644 + 473.1844 + 0.2724 = 3438.25 \]
Then we can find \(s\):
\[ s = \sqrt{\frac{3438.25}{10 - 1}} = \sqrt{\frac{3438.25}{9}} = \sqrt{382.0278} \approx 19.55 \]
Step 3: Determine the sample size
The sample size \(n = 10\).
Step 4: Determine the critical value \(t^*\)
Using the t-distribution for 95% confidence level and \(n - 1 = 9\) degrees of freedom, we can find \(t^*\) from a t-table or calculator.
For \(df = 9\):
\(t^* \approx 2.262\) (rounded to three decimal places).
Step 5: Calculate the margin of error
The margin of error (ME) is given by:
\[ ME = t^* \times \frac{s}{\sqrt{n}} \]
Calculating the margin of error:
\[ ME = 2.262 \times \frac{19.55}{\sqrt{10}} \approx 2.262 \times \frac{19.55}{3.162} \approx 2.262 \times 6.182 \approx 13.96 \]
Step 6: Construct the confidence interval
To find the confidence interval, we calculate:
\[ \bar{x} - ME \quad \text{and} \quad \bar{x} + ME \]
\[ C.I. = (68.38 - 13.96, 68.38 + 13.96) = (54.42, 82.34) \]
Thus, the final answer for the 95% confidence interval is:
\[ \text{95% C.I.} = (54.42, 82.34) \]
So, the answer is:
\[ \text{95% C.I.} = (54.42, 82.34) \]